Let fk(x) = (1/k)(sin(kx) + cos(kx)) where x ∈ R and k ≥ 1. Then f4(x) - f6(x) equals

f4(x) = (1/4)(sin(4x) + cos(4x))
f6(x) = (1/6)(sin(6x) + cos(6x))
f4(x) - f6(x) = (1/4)(sin(4x) + cos(4x)) - (1/6)(sin(6x) + cos(6x))
This expression cannot be simplified to a constant value without further information or constraints on x. However, if the question intended to ask for the difference in the coefficients of the trigonometric terms rather than the function values, then we can proceed as follows:
Let's consider the case when x=0:
f4(0) = (1/4)(sin(0) + cos(0)) = 1/4
f6(0) = (1/6)(sin(0) + cos(0)) = 1/6
f4(0) - f6(0) = 1/4 - 1/6 = (3-2)/12 = 1/12
This does not match any of the given options. Let's check if there's a calculation error in the original provided solution:
The provided solution seems to contain errors and is not directly related to the question. There is no valid simplification to reach 1/12 or any of the options provided. The problem likely needs a correction in its statement or options.